d²x = u + vx + w dx ,
dt² dt

where u, v, w are functions of t, by writing y for dx/dt, is equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In fact a similar reduction is possible for any system of differential equations with one independent variable.

Equations occur to be integrated of the form

Xdx + Ydy + Zdz = 0,

where X, Y, Z are functions of x, y, z. We consider only the case in which there exists an equation φ(x, y, z) = C whose differential

∂φ dx + ∂φdy + ∂φ dz = 0
∂x ∂y ∂z

is equivalent with the given differential equation; that is, μ being a proper function of x, y, z, we assume that there exist equations

∂φ = μX, ∂φ= μY, ∂φ= μZ;
∂x ∂y ∂z

these equations require

∂ (μY) ≈ ∂(μZ), &c.,
∂z ∂y